on ludvig lorenz and his 1890 treatise on light …jerf/papers/on_ll.pdfthree textbooks on,...

The European Physical Journal HThis is a post-peer-review, pre-copyedit version of an article published in EPJ H.The final authenticated version is available online athttps://doi.org/10.1140/epjh/e2019-100022-y

On Ludvig Lorenz and his 1890 treatise onlight scattering by spheres

Jeppe Revall Frisvad1,a and Helge Kragh2,b

1 Department of Applied Mathematics and Computer Science, Technical University of Den-mark, Richard Petersens Plads, Building 321, 2800 Kongens Lyngby, Denmark

2 Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Den-mark

Abstract. This paper offers background and perspective on a little-known memoir by Ludvig Lorenz on light scattering by spheres, whichwas published in Danish in 1890. It is a companion to an English trans-lation of the memoir appearing separately. Apart from introducingLorenz and some of his contributions to optics and electrodynamics,the paper focuses on the emergence, content and reception of the 1890memoir and its role in what is often called the Lorenz-Mie theory.In addition to the historical analysis, the paper illuminates aspects ofmodern Lorenz-Mie theory and its many applications, with an eye toLorenz’s original work.

1 Introduction

According to Google Scholar, the term “Lorenz-Mie” appears in about 7,300 scien-tific articles of which 5,900 date from the last two decades. While the second namerefers to Gustav Mie, a well-known German physicist who published his theory in1908, the first name is much less known. In a nutshell, in a lengthy memoir of1890 the Danish physicist Ludvig Lorenz published an elaborate optical theory onthe scattering of light by small transparent spheres, which was essentially a non-electromagnetic version of Mie’s later theory. Mie’s seminal paper was written inGerman and originally it attracted very little attention. Reflecting the growing in-terest in Mie’s theory, the paper has been translated into English and thus madeaccessible to physicists and historians of physics all over the world [Wriedt 2012,p. 55; https://scattport.org/index.php/classic-papers]. Lorenz published his memoirin Danish, which effectively has made it unknown to the physics community. Onlynow, almost 130 years later, we can present a complete English translation of thisgreat and complex paper which has been called “one of the most remarkable memoirsto be published in the 19th century” [Logan 1965, p. 77].

In the present paper, a companion to the translation, we discuss the historicalcontext of Lorenz’s scattering theory and relate it to some of his earlier contributionsto optics and electrodynamics. We also include a brief biographical section and outline

a e-mail: [email protected] e-mail: [email protected]

https://doi.org/10.1140/epjh/e2019-100022-y

https://scattport.org/index.php/classic-papers

2 The European Physical Journal H

the main trajectory that led to Mie’s celebrated theory of 1908. From a historical pointof view the reception of a scientific work is no less important than the work itself, andfor this reason we refer to how Lorenz’s theory was received by contemporary andlater physicists. The second part of our paper is less historically oriented as it takesup in a more critical way some of the mathematical and physical problems appearingin or related to Lorenz’s memoir. Moreover, in this part we also discuss, if only brieflyand selectively, the revival of interest in Lorenz-Mie theory and aspects of its currentsignificance and applications.

2 Biographical sketch

Ludvig Valentin Lorenz (1829-1891) was born in Elsinore, the city housing the castleof Shakespeare’s Hamlet. “Very early on I showed an interest in calculations and math-ematics,” he recalled in an autobiographical note of 1877 [Kragh 2018b, pp. 243–245].However, rather than studying mathematics or physics, as a young man he enteredthe Polytechnic College in Copenhagen to study applied or engineering chemistry.During his studies he followed a course in chemical physics given by the aging HansChristian Ørsted, whose romantic natural philosophy influenced him to some degree.On the other hand, he generally preferred – contrary to Ørsted – a mathematicalrather than a philosophical approach to physics.

Lorenz graduated in 1852 without distinction and without being seriously inter-ested in his chosen field of study. Indeed, none of his later scientific works related toproblems of a chemical nature. During most of the period 1852-1866 Lorenz lived amodest life as part-time teacher at middle and gymnasium schools in Copenhagen,where he taught elementary physics, chemistry and mathematics. At the same timehe continued his private studies of still more advanced topics in theoretical physics. Itwas a turning point in Lorenz’s life when he, in 1858, received grants that enabled himto travel to Paris and study physics at Sorbonne University. Attending lecture coursesby illustrious physicists such as Joseph Liouville, Gabriel Lame and Henri Regnaulthe was for the first time confronted with modern and advanced mathematical physics.As a direct result of his one-year stay in Paris, in 1860 he published his first scientificpaper on theoretical optics. Two years later he married Agatha Fogtmann, who wasto survive her husband by 31 years. The marriage was childless.

Lorenz’s fortunes changed to the better only in 1866, when he obtained for thefirst time a permanent position as physics teacher at the Royal Military High Schoolestablished in 1829. In this position, he had at his disposal an excellent laboratoryand instrument collection which allowed him to engage in experimental research andnot only theoretical physics. For his teaching at the Military High School, he wrotethree textbooks on, respectively, general physics, optics, and heat. The book on opticswas translated into German and published by the recognized Leipzig publisher B. G.Teubner [Lorenz 1877]. In the same year that Lorenz was appointed physics teacherat the Military High School, he was elected a member, at the relatively young ageof 37, of the prestigious Royal Danish Academy of Sciences and Letters founded in1742. Although Lorenz never became a professor and never wrote a doctoral disserta-tion, he came to be recognized as an important member of the small Danish physicscommunity and the country’s first and foremost theoretical physicist (Figure 1). Ina testimony of 1873 to the Ministry of Church and Education, the German-bornprofessor of astronomy Heinrich Louis d’Arrest wrote as follows: “I do not hesitateto point out to the high Ministry that associate professor Lorenz is one of the firstand internationally respected scientists within this field [mathematical physics]. Hisextremely penetrating investigations belong to the most difficult and remarkable ac-complishments of our time” [Kragh 2018b, p. 238].

J.R. Frisvad and H. Kragh: On Ludvig Lorenz and his 1890 treatise 3

Fig. 1. Ludvig V. Lorenz. Royal Library, Copenhagen, Picture Collection.

Compared to other Danish physicists in the period Lorenz was internationallyoriented and for several decades the only one with a solid reputation abroad. He wasprobably better known and more appreciated by physicists in German-speaking Eu-rope than in his own country. Not only did he publish in leading physics journals suchas Journal de Physique, Crelles Journal and Annalen der Physik und Chemie, he alsocorresponded with several of Europe’s foremost physicists. Among his correspondentswere distinguished scientists including Ludwig Boltzmann, Carl Neumann, FriedrichKohlrausch, Heinrich F. Weber, August Kundt and, in France, the later Nobel Prizelaureate Gabriel Lippmann. He sent regularly off-prints of his papers to these andother physicists abroad, including notables such as Anders Angstrom, Marcel Bril-louin, Henri Poincare, Janne Rydberg and Hermann von Helmholtz. On the otherhand, he had practically no contact to British physicists, most conspicuously not toJames Clerk Maxwell and Lord Rayleigh.

While quite satisfied with his position at the Military High School, in early 1887Lorenz received a generous offer that he could not decline. The Carlsberg Foundationestablished by the wealthy brewer Jacob C. Jacobsen in 1876 offered him a life-longsalary as an independent scientist who could cultivate whatever field of study he foundsuitable. Lorenz happily accepted the offer, which meant that he had to resign hisposition at the Military High School and therefore also quit experimental research.During the brief span of years in which he worked as a “free scientist” he completedhis ambitious analysis of the scattering of plane waves by spherical bodies aboutwhich more below. After this tour de force he focused on a work in pure mathematicsdealing with prime numbers, which was to be his last publication.

Lorenz contemplated a future work on the famous three-body problem which atthe time attracted much mathematical attention. However, he did not come very farin his study of the intricate problem. On 9 June 1891 he died unexpectedly of a heartattack, at the age of 62. While the memory of Lorenz soon faded, internationally aswell as in his native country, a few years after his death the Carlsberg Foundationdecided to honour him with a French edition of his collected papers. The result wasOeuvres Scientifiques de L. Lorenz edited by the mathematician Herman Valentiner


and published in two volumes around the turn of the century [Lorenz 1904]. Thefirst of the volumes, today readily available online on https://archive.org/details/oeuvresscientifi01lore, includes the 1890 memoir.

3 An electrical theory of light

Whereas Lorenz’s early optical theories were within the established tradition of theelastic theory of light, in 1867 he proposed a new theory in which light waves were con-sidered to be oscillating electrical currents and not vibrations in an ethereal medium.As he wrote, there was “scarcely any reason for adhering to the hypothesis of anether, for it may well be assumed that in the so-called vacuum there is sufficientmatter to form an adequate substratum for the motion” [Lorenz 1867]. The theorywas in part inspired by the dynamical ideas of Ørsted, his former teacher, which hedeveloped into a concise mathematical theory. Building on an earlier theory of thepropagation of electricity in conducting media proposed by Gustav Robert Kirchhoff[1857], but unaware of Maxwell’s recent electromagnetic theory of light, Lorenz ar-rived at equations for the variation of the current density j. In a condensed form andwith σ denoting the electrical conductivity, his wave equation for j can be expressedas

−∇× (∇× j) =1

a2∂2j

∂t2+

16π

a2σ∂j

∂t,

where a is the velocity of the electrical wave (or light wave). Lorenz identified thelast term with the absorption of light which would increase with the conductivity. Itis to be noted that there were no electrical or magnetic fields in the 1867 theory andnor were there any displacement currents. All electrical currents and hence also lightwere conduction currents.

Although Lorenz’s electrodynamic theory of light has long ago been abandoned,two of the important results of the 1867 paper are still parts of modern physics. Lorenzemphasized that the action of electrical disturbances requires time to propagate cor-responding to a finite velocity as given by retarded potentials. Although the generalidea of retarded electrical action had previously been suggested (by C. Friedrich Gaussand Bernhard Riemann, see Kaiser [1981]), Lorenz was the first to incorporate theidea into a coherent theory of light and electricity. For a point x = (x, y, z) in avolume V , his formula for the scalar potential due to a charge density ρ was

ϕ(x, t) =

∫ρ(x′, t− r

a )

rdV,

where r = |x − x′| and dV = dx′ dy′ dz′. Similarly, Lorenz’s formula for the vectorpotential A was

A(x, t) =

∫j(x′, t− r

a )

rdV .

The other technical innovation appearing in Lorenz’s paper concerned the relationshipbetween the retarded potentials. Following Kirchhoff’s notation, he originally used Ωfor the scalar potential and (α, β, γ) for the vector potential, arguing that they hadto satisfy the constraint

∂Ω

∂t= −2

(∂α

∂x+∂β

∂x+∂γ

∂x

).

In modern notation and Gaussian units, the factor 2 disappears and the constraintbecomes

∇ ·A +1

c

∂ϕ

∂t= 0 .

https://archive.org/details/oeuvresscientifi01lore

https://archive.org/details/oeuvresscientifi01lore


This is the famous Lorenz condition, by far the most useful of the various gaugesused in electrodynamics. Until recently the gauge stated in 1867 was predominantlynamed the “Lorentz gauge” because Hendrik A. Lorentz used it in his importantwork on electromagnetic theory in the early years of the twentieth century. Contraryto the Coulomb gauge ∇·A = 0, this gauge is manifestly Lorentz invariant. Although“Lorentz gauge” is still widely used, paternity belongs to Lorenz of Copenhagen [Jack-son and Okun 2001].

Lorenz’s electrical theory of light was well known to contemporary physicistsand referred to by, for example, Maxwell, Heinrich Hertz, Eleuthere Mascart, CarlFriedrich Zollner, and Oliver Lodge. In the early part of the twentieth century it wascritically discussed by Pierre Duhem. However, by that time it had largely fallen intooblivion. We shall not here examine the fate of Lorenz’s electrical oscillation theory oflight, for which we refer to the recent literature [Keller 2002; Kragh 2018b,c]. An im-portant reason for the cool reception and later neglect was undoubtedly that Lorenzdid not follow up upon it or refer to it in his later works. Nor did he ever refer toMaxwell’s field theory of light and its relationship to his own. It is remarkable thatthere is no trace of electromagnetism in Lorenz’s great work on the scattering of lightby spheres. The memoir of 1890 contains no reference to his earlier theory and wordssuch as “electrical” and “magnetic” do nowhere appear significantly.

4 Optical theories, ca. 1860–1883

Influenced by his research stay in Paris, Lorenz’s first scientific papers were based onthe elastic or mechanical theory of light which still in the early 1860s constituted thegenerally accepted theoretical framework in optics. According to this framework orparadigm, light consisted of transverse waves made up of deformations in an incom-pressible ethereal medium [Whittaker 1958, pp. 128–170; Darrigol 2012, pp. 225–239].About 1860, when Lorenz entered the field, the incompressibility of the solid ether andthe transverse nature of the vibrations therein were subjects of debate which Lorenz’searly works in optics helped to clarify. Most of these works appeared in Annalen andwere reviewed in Fortschritte der Physik, the abstract journal published since 1845by the German Physical Society. They were thus well known to the internationalcommunity of physicists.

With the purpose of understanding the ether vibrations in the reflection and re-fraction of light, Lorenz [1861] assumed that the two optical media were separatedby a boundary or transition layer. This layer he imagined to be divided in an infinitenumber of infinitesimally thin sheets, each of the sheets having a constant density.He estimated that the thickness of a typical layer was between 1 and 10 per cent ofthe wavelength of visible light or roughly between 5 nm and 50 nm. Lorenz’s conceptof an optical transition layer was original and later recognized to be an importantinnovation, but at the time it was either criticized or ignored [Schuster 1909, p. 241;see also Keller 2002]. The notion of a continuum of transition layers reappeared inLorenz’s later scattering theory, now adapted to spherical surfaces. With ω denotingthe velocity of the light, the hypothesis entered as follows: “This discontinuous transi-tion is considered to be produced by a surface layer of finite thickness and continuouschange of ω . . . that approaches a layer of thickness zero” [Lorenz 1890, pp. 4–5].

After less than two years of work within the mechanical-elastic paradigm, Lorenzrealized that the laws of optics could not be fully deduced from the theory of elasticity.Adopting an alternative non-mechanical methodology, he now concluded that thefundamental theory of light must rest solely on abstract conceptions in agreement withobserved phenomena. The “phenomenological” approach implied that the physicalnature of light was disregarded and also that the ether was given less attention.


Although the ether still figured in Lorenz’s papers until 1865, it was merely as aname without physical characteristics. In a paper of 1863, he introduced an abstractlight vector u = (ux, uy, uz) on the basis of which he formulated three differentialequations [Lorenz 1863]. He claimed that in heterogeneous, non-absorbing media allphenomena of light can be deduced from these equations, which in modern notationcan be written as

−∇× (∇× u) =1

a2∂2u

∂t2. (1)

Lorenz had a = ω/k, where ω is the angular frequency and k is the wave number. Thisis equivalent to a = c/n, where c is the speed of light in vacuo and n is the index ofrefraction. The same differential equations entered into Lorenz’s 1867 electrical theoryof light, except for the added absorption term in the latter theory. The equations alsoformed the basis of his 1890 scattering theory. “The differential equations from whichthe present investigation takes its starting point,” Lorenz stated in his memoir (p. 3),“differ from the theory of elasticity by the fact that they rule out the possibility oflongitudinal oscillations, and, since they are valid for every point in any transparentheterogeneous medium, the boundary conditions at the transition from one bodyto another can be derived from the differential equations themselves.” It should bementioned that much earlier the Irish physicist James MacCullagh had derived a waveequation similar to Lorenz’s, but MacCullagh’s theory rested on the assumption ofan elastic solid ether [Darrigol 2010].

In a paper of 1869, Lorenz derived on the basis of experiments and his opticalwave theory a relationship between a transparent body’s refractivity index N and itsspecific volume v as given by v = 1/d, where d is the density. He concluded that

N2 − 1

N2 + 2v = constant .

For reasons of simplicity, he assumed the refractive medium to be composed of opti-cally homogeneous spherical molecules with Ni being their internal refractive index.With vi being the specific proper volume of the molecules, Lorenz could then statethe law as

N2 − 1

N2 + 2v =

N2i − 1

N2i + 2

vi .

Nine years later, 25-year-old H. A. Lorentz, who had recently been appointed pro-fessor at the University of Leiden, derived the same refractivity formula but on thebasis of the electromagnetic theories of Maxwell and Helmholtz. Since Lorentz’s pa-per was in Dutch, and Lorenz’s in Danish, the “Lorentz-Lorenz law” became widelyknown only after 1880, when revised versions of the two memoirs appeared in Germantranslations [Lorentz 1880; Lorenz 1880]. As to Lorenz, he used his refraction-densitytheory to estimate a lower limit to the size of molecules. For what he called “theradius of the molecular sphere of action,” a quantity greater than the actual radius ofthe molecule, he found its lower limit to be 15 nm. Lorenz returned to the question ofthe size of molecules in the last sections of his 1890 memoir. More information aboutthe early history of what is known as either the Lorentz-Lorenz or the Lorenz-Lorentzlaw can be found in Kragh [2018a].

Finally, among the sources for Lorenz’s scattering theory of 1890 was also anextensive paper on chromatic dispersion published in Annalen [Lorenz 1883; Keller2002, pp. 283-285]. As Lorenz [1890, p. 3] pointed out in his introduction, this pa-per served as the direct inspiration for his more elaborate theory of the scattering ofplane light waves by a transparent and isotropic sphere. Moreover, it contained math-ematical concepts and methods that would appear in his memoir seven years later.Although the basic equations of the 1890 treatise were the same as those employed


in his dispersion theory of 1883, Lorenz [1890, p. 3] now “preferred a different andsimpler way of presentation where I, also to ease the reading, shall avoid assumingknowledge of my previous work.”

The 1883 dispersion theory assumed a collection of randomly ordered point atomssurrounded by concentric shells. Each shell was characterised by a constant refractiveindex, which diminished with its distance from the atom, corresponding to a variationof the effective velocity of light. Based on this model Lorenz derived a dispersionrelation, but unfortunately it was unable to account for anomalous dispersion. Forthis and other reasons his dispersion relation was considered to be unsatisfactory.

5 From Clebsch to Mie

Lorenz presented his new scattering theory to the Royal Danish Academy in an ad-dress of 18 October 1889 in which he called attention to an earlier work by the Germanmathematician and physicist Adolf Clebsch:

I need to mention a single attempt to determine the reflection of light from aperfectly bright spherical surface . . . been made by Clebsch but with the deter-rent results, as he puts it in the explanations to his great work (Crelles Journalof 1863), that the results of the entire investigation are terribly complex. . . . Onthe other hand, he has succeeded in the case of very small reflecting spheres.The mentioned author proceeds from the differential equations of elasticitytheory, [while] . . . I have myself based the theory on different equations, thesame which I proposed for approximately 25 years ago. [Kragh 2018b, p. 104]

Clebsch died in 1872, only 46 years old. As assistant professor at Karlsruhe Poly-technic College he played an active role in the German mathematical community,contributing with innovative ideas in both pure mathematics and theoretical physics.For example, in 1866 he wrote a mathematical treatise on Abelian functions jointlywith Paul Gordan. The Clebsch-Gordan coefficients used extensively in quantum me-chanics and elementary particle physics have their roots in this treatise. More tothe point, what caught Lorenz’s attention was a mathematically challenging 68-pagememoir on the reflection of elastic waves by a spherical obstacle [Clebsch 1863; Tod-hunter 1893, Part 2, pp. 168–180; Logan 1965]. Although published in one of the pe-riod’s most important mathematical journals, Crelles Journal, Clebsch’s pre-Maxwellmemoir exerted almost no interest at all on either the mathematicians or the physi-cists. In fact, during the late nineteenth century Lorenz was alone in recognizing itsimportance and receiving inspiration from it. However, since Lorenz’s Danish treatiseremained practically unknown, its highlighting of Clebsch’s work made no difference.

Apart from Clebsch’s 1863 memoir, Lorenz also cited earlier works by GeorgeStokes, George Airy, Lord Rayleigh (or John William Strutt) and a few other authors.Of Rayleigh’s extensive writings on the scattering of waves he only cited an early setof papers based on the equations of elastic waves in which Rayleigh [1871a,b,c] hadderived the scattering law for small particles. Although Lorenz was presumably awareof Rayleigh’s work of 1881 [Kerker 1969], which offered an electromagnetic derivationof the scattering law, he did not refer to this work and also not to other works basedon Maxwell’s theory of light. At the end of his memoir, Lorenz considered the case ofsmall spheres of molecular dimensions. He proved that the scattering cross section ofa single sphere of radius R very small compared to the wavelength λ is

Cs =128

3

π5R6

λ4

(N2 − 1

N2 + 2

)2.


This equation provides both the scattering cross section of a small particle and its linkto the polarizability of a spherical particle as given by the Lorentz-Lorenz relation.Since the refractive index for air and other transparent bodies only varies slightlyin the wavelength range of visible light, this expression for Cs is dominated by theλ−4 term. As Lorenz pointed out, this result agreed with what Rayleigh had foundin 1871. For N close to 1, it gives approximately the same expression as found byRayleigh, namely

Cs =128

27

π5R6

λ4(N2 − 1)2 .

In his later theory of scattering, Rayleigh [1899] explains that his original scatteringcross section is only valid for N close to 1 as the shape of the particle must be takeninto account in the preferred electric theory [Rayleigh 1881]. He continues to say thatN2 − 1 should be replaced by 3(N2 − 1)/(N2 + 2) in the case of spheres. The twotheories then agree very well. The remaining difference between them is Rayleigh’s[1871a] simple expression for the distribution of the scattered light (the Rayleighphase function), where Lorenz has the fully general solution for a sphere. Rayleigh’stheory is in other words for particles significantly smaller than the wavelength withspheres as a special case, whereas Lorenz’s theory is for spheres of arbitrary size withsmall spheres as a special case.

Rayleigh [1899] also derived an expression for the number density of scatteringparticles Np in terms of the refractive index N of the gas in question:

Np =32π3

3hλ4(N − 1)2 ,

Here, h is the extinction coefficient (scattering plus absorption), which is an observablequantity. For non-absorbing particles, the extinction coefficient is only a measure ofthe amount of scattering per unit of distance as light travels through the medium, andthen h = CsNp. Lorenz referred to h as the absorption coefficient. For an ideal gasunder standard conditions, the molecules become the scatterers andNp corresponds toLoschmidt’s number NL. Loschmidt’s number relates to Avogadro’s number through

NA = 6.022 · 1023 mol−1 ≈ NL · 22.4 · 103 cm3 mol−1 .

Lorenz’s formula as given in the 1890 memoir was

Np =24π3

hλ4

(N2 − 1

N2 + 2

)2.

If N is only slightly larger than one, which is certainly the case for a gas, Lorenz’sexpression is fairly similar to the one stated by Rayleigh nine years later. Lorenzillustrated his theory by considering a concrete example where plane waves of lightof wavelength λ = 580 nm pass through atmospheric air. In this case, he foundthe value Np = 1.63 · 1019 cm−3, which is more than half of NL. For the case ofspherical particles, several of the results communicated by Rayleigh in 1899 can befound in Lorenz’s earlier work. Considering the historical development of the theoryof scattering by a sphere, Milton Kerker [1969, p. 59] concludes that “certainly ifthis theory is to be associated with the name or names of individuals, at least thatof Lorenz, in whose paper are to be found the practical formulas so commonly usedtoday, should not be omitted.” Rayleigh only referred to Lorenz’s theory in 1918, inthe autumn of his life, and then only briefly.

Contrary to Lorenz, when Gustav Mie in 1908 published his now very famoustreatise on the scattering by conducting spheres, he did not refer to Clebsch’s memoir.


Fig. 2. Number of citations to G. Mie’s 1908 paper as a function of time according to Webof Science.

Nor did he cite Lorenz’s paper in either its Danish original or the French translationavailable since 1898. Mie’s theory has recently been the subject of historical attentionand in the present context there is no need to go into details, as these can be foundin the literature [Horvath 2009; Hergert 2012; Wriedt 2012]. At the time professorat the University of Greifswald, Mie was an expert in Maxwellian electrodynamicsand a leading figure in the ill-fated, pre-Einsteinian attempts to establish this theoryas the foundation of all of physics [Vizgin 1994]. Today he may best be known forhis 1908 scattering theory, which more specifically was a theory of colour effectsof colloidal metal solutions, but he did not himself consider it to be particularlyimportant. Nor did most other physicists, for originally it only attracted modestattention such as indicated by the very few references to it until about 1970 (Figure 2).What matters here is the relation of Mie’s paper to Lorenz’s, which today is widelyseen as an anticipation of it or even an equivalent but physically different version ofMie’s electromagnetic theory.

Although Mie did not formally cite Lorenz’s paper on scattering by a sphere, hecited Lorenz [1880] on the refractivity-density law and may even have known of his1890 memoir if without studying it in any detail [Kragh 2018b, pp. 115–116]. What-ever the historical connection between the two papers, today it is widely recognizedthat they are empirically equivalent. As part of his Munich doctoral dissertation Pe-ter Debye [1909a] wrote an important paper on the scattering problem, including ananalysis of the light pressure exerted on a sphere, in which he cited the earlier papersof Clebsch and Mie. Although he did not refer to Lorenz’s memoir, he was aware ofit and had studied it in its French translation. This is evidenced by another paperof a predominantly mathematical nature in which Debye [1909b] referred to formulaefirst stated by Lorenz.

While Lorenz’s theory was abstract and formulated without a physical interpre-tation of light and its propagation through space, Mie’s was solidly founded on theMaxwell-Lorentz theory of electromagnetism. According to Gouesbet [2012, p. 74],Lorenz’s scattering theory rested on the assumption of the ether, while Mie’s didnot. Horvath [2009] likewise states that “Lorenz completely solved for the scatteringof light by small particles theoretically using the ether theory.” However, this is amisunderstanding. It was Mie’s electromagnetic theory, not Lorenz’s, which relied onthe ether, and it did so because it was based on the Maxwell-Lorentz ether theory.The ether played no role at all in Lorenz’s memoir, where the word “ether” simplydid not appear.

Lorenz made his theory of light “free from all physical hypotheses” to preventthat the mathematical model would be nullified by “further progress of a future


time” [Lorenz 1867]. This turned out to be a good decision as his solution for thescattering of light by a sphere from 1890 is mathematically equivalent to those of Mieand Debye. The reason is that, for a non-magnetic substance (permeability equal tounity), the exact same wave equation as the one used by Lorenz (1) can be derivedfor the electric vector using Maxwell’s equations [Born and Wolf 1999, p. 11]. More-over, for a plane wave and a spherical wave, the magnetic vector is not required. Weonly need the electric vector and the direction of wave propagation to determine thePoynting vector and thus the distribution of radiant energy. In other words, Lorenzselected the exactly right phenomenological model for his theoretical investigations,and we can therefore directly use his results even today.

The term Lorenz-Mie theory as a substitute for Mie theory first turned up inthe early 1960s [Wait 1962; Logan 1962]. As stated by Wait [1962], “The theory ofscattering of electromagnetic waves from a homogeneous sphere was first given inexplicit form by Lorenz and some time later by Mie.” Strictly speaking this is ofcourse incorrect, since Lorenz’s theory was not about electromagnetic waves. Despitethe increased recognition of Lorenz’s contribution, it is still more common to speak ofthe Mie theory, whereas Mie-Lorenz theory is rarely used. According to Google Scholara total of ca. 92,300 papers refer to Mie theory and ca. 4,930 to Lorenz-Mie theory.For the period 1980–2018 the corresponding numbers are ca. 42,700 and ca. 4,780.Out of these 4,780 scientific articles, around one third are referring to generalizedLorenz-Mie theory (GLMT, more on this theory later). GLMT has thus been quiteinfluential with respect to inclusion of Lorenz’s name.

6 Contents of Lorenz’s memoir in brief

Lorenz started his 1890 memoir by referring to Clebsch [1863] and to his own previouswork [Lorenz 1883]. He also set the scene by considering the differential equations (1)of his previous work the model of light propagation (the variable we denote by a isdenoted by ω in Lorenz’s papers). All his results were derived from this model of lightpropagation. Throughout the memoir, Lorenz specifically considered the interaction ofa plane wave with a spherical surface. Like Mie [1908] and Debye [1909a] after him, hesolved the differential equations describing the light propagation by reformulating theproblem in spherical coordinates. The incident plane wave was expanded in sphericalharmonics (using a formula first found by Bauer [1826]), and the boundary conditionswere used to obtain the spherical harmonics expansion coefficients of the scatteredwave. Numerical evaluation of the solution was difficult, so Lorenz spent many pagesdeveloping a theory for approximate evaluation of special cases.

The size parameters for a particle of radius R and relative index of refraction Nare

x =2πR

λand y =

2πRN

λ,

where λ is the wavelength in vacuo. Lorenz used α and α′ to denote x and y andconsidered the special cases of α very large and α very small, that is, large andsmall particles as compared with the wavelength. He also included a section on thespecial case of light getting trapped in a thin layer inside the sphere due to totalinternal reflection (when N < 1). Lorenz used the case of α very large to discuss theappearance of rainbows and the case of α very small to discuss absorption lines insystems of particle and similarities of his theory with that of Rayleigh.

Lorenz considered the right mathematical model (as explained in the previoussection) and was therefore able to derive accurate formulae for the amount and theintensity of the scattered light. However, since he did not offer a physical hypothesis,he could not draw electric and magnetic field lines of partial waves like Mie did. Mie


[1908] solved the differential equations separately for the electric and magnetic vectors.Then, based on Poynting’s theorem [Poynting 1884], he found that the solution for theelectric vector was sufficient “to obtain a clear representation of the radiation.” Basedon Mie’s observations, Debye [1909a] showed that we only need to solve differentialequations for a pair of scalar potentials. From these, we can calculate the full solutionfor the electric and magnetic vectors outside the sphere. Although Lorenz consideredthe wave vector corresponding to the electric vector only, he found the same pair ofscalar potentials. In addition, he had a pair of potentials relating to the scatteredwave inside the sphere.

Interestingly, Mie [1908] mentioned that his theory could be used for the rainbowproblem. However, he found that this would require taking into account a ratherlarge number of partial waves, which would result in “considerable difficulties incalculation.” This explains Lorenz’s need for lengthy calculations to approximate thecase of α very large.

7 Context and reception of the 1890 theory

Lorenz’s great work was published in Danish in the not widely circulated transactionsof the Royal Danish academy and for this reason alone it remained unknown or atleast inaccessible to the international community of physicists. At the turn of thecentury it was available in French translation, but Oeuvres Scientifiques de L. Lorenzwas rarely read and failed to make the theory more generally known. Although thelanguage barrier was undoubtedly a major reason for its lack of visibility, it was noless important that the theory ignored what at the time was considered modern fieldelectrodynamics. By the late 1890s Maxwell’s ether-based theory of electromagnetismcompletely dominated physicists’ thinking about optics, so why pay attention to acomplex and obscure theory of the past that failed to take into account electromag-netic fields and the associated ether? Kerker [1969, p. 56] ascribes the neglect ofLorenz’s theory to be in part that “His solution is based upon his own theory of elec-tromagnetism rather than on that of Maxwell.” The word “electromagnetism” shouldprobably have been “light propagation” in this sentence since it was Lorenz’s 1863theory of light propagation rather than his 1867 electrical theory of light that playeda role in his later scattering theory.

In spite of the language barrier and the non-electromagnetic framework of Lorenz’sscattering theory, it was not entirely neglected. For the few early references to thetheory, see Logan [1965] and Kragh [2018b, pp. 112–115]. It is worth noting that theBritish physicist J. C. Maxwell Garnett [1904] in an important paper on the colourof metals cited Lorenz’s Danish memoir and that Mie four years later cited MaxwellGarnett. By the way, Maxwell Garnett (1880-1958) was not related to the famous J.C. Maxwell although he was named after him. Also worth noting is that H. A. Lorentznever mentioned Lorenz’s 1890 theory or worked on the optical scattering problem.Nonetheless, in a textbook of atmospheric optics we read that Mie’s theory “wasanticipated by Lorentz during the period 1890 to 1900” [McCartney 1976, p. 217].The author possibly had Lorenz in mind but may have confused him with the famousDutch physicist, such as was commonly done (and is still done) in connection withother of Lorenz’s contributions and with the Loren(t)z gauge condition in particular.It is more than a little disturbing to find in the physics literature about 740 referencesto the non-existing “Lorentz-Mie theory.”

Throughout his life Lorenz nourished a burning interest in mathematics whetherof the pure form or as applied to problems of physics [Kragh 2018b, pp. 207–215].Indeed, to the limited extent that Lorenz’s work of 1890 was known in the early partof the twentieth century it was more because of its mathematical rather than its


physical content. When Debye in 1909 cited Lorenz’s memoir it was because of themathematical formulae it contained, and the same was the case with several otherciting authors. Harry Bateman, a talented mathematician and theoretical physicist,was aware of Lorenz’s memoir to which he referred in a book on electrical and opticalwaves. As Bateman [1915, p. 79] noted, for the case of a spherical body Lorenz had“long ago” studied the diffraction problem in relation to the rainbow. Some years ear-lier Bateman’s compatriot, the mathematical physicist John Nicholson [1910], studiedexact solutions of how waves are scattered by a sphere, and in this connection he dis-cussed asymptotic formulae for Bessel functions with reference to results found inLorenz’s 1890 memoir. In this paper, Lorenz had derived a formula for the asymp-totic expansion of products of Bessel functions of the form J2

α + Y 2α , where the two

symbols refer to Bessel functions of the first and the second kind. The result attractedthe attention of the Cambridge mathematician George N. Watson, who in his mon-umental treatise on Bessel functions credited Lorenz with the discovery of a formulawhich previously had been thought to date from 1906 [Watson 1922, p. 22 and p. 29].Lorenz’s expansion formula appears as Eq. (66) in his 1890 memoir and the quanti-ties introduced by him are currently known under the name Lorenz-Mie coefficients.Logan [1965], Kerker [1969], and Wait [1998] refer to other mathematical innovationsappearing in Lorenz’s “lost memoir.”

8 Lorenz-Mie theory then and now

The usefulness of an analytic solution for an idealized special case like the scatteringof a plane wave by a perfect sphere is remarkable. When developing more generalscattering theory, it is good practice to validate against limiting cases such as a perfectsphere or a plane wave. This is one of the key reasons why Lorenz’s work of 1890 hasremarkable longevity and is still being cited more than five quarter-centuries afterits publication. Another key reason behind the longevity of the Lorenz-Mie theoryis its ability to predict scattering effects when particles are somewhat like spheres.Prediction of rainbow effects [Lorenz 1890] and the brilliant colours of colloidal metalsolutions [Mie 1908] are just examples. A review by Kerker [1982] illustrates therichness of the Lorenz-Mie theory in explaining scattering phenomena. At this pointin time (1982), almost a century after Lorenz published his work, Kerker points outthat “within the lode of the Lorenz-Mie equations [. . . ] most of the treasures stillto be mined have not yet been brought to view.” Even now, the Lorenz-Mie theoryseems to hold hidden treasures.

In various parts of Lorenz’s 1890 treatise, his conclusive equations are surprisinglyclose to the equations appearing in modern articles and textbooks on scattering ofelectromagnetic radiation (or light). Let us consider Lorenz’s exposition of the Lorenz-Mie coefficients, which reappear again and again in generalized versions of the theory.Naming conventions for special functions were of course not quite as firmly settledin the late 19th century, so Lorenz provided mathematical definitions. For example,Lorenz defined Legendre polynomials (Pn) using a recurrence relation (last equationof page 8) and used them for expansion just like in a modern text book [see Arfkenet al. 2013, for example]. As in most later expositions of Lorenz-Mie theory, Lorenzexpanded the complex exponential appearing in the wave equations and found that avariation of Bessel functions provides a solution for the scattered wave. The variationis today called Riccati-Bessel functions ψn and ζn, and these are typically defined interms of spherical Bessel functions jn and yn. Lorenz used vn and wn, which we canconnect to the Riccati-Bessel and spherical Bessel functions as follows:

vn(z) = ψn(z) = zjn(z) and vn(z) + wn(z) i = ζn(z) = zjn(z)− i zyn(z) .


Lorenz’s equations (33) for the Lorenz-Mie coefficients are

2kn = −1− (vn(α)− wn(α) i) v′n(α′)−N (v′n(α)− w′n(α) i) vn(α′)

(vn(α) + wn(α) i) v′n(α′)−N (v′n(α) + w′n(α) i) vn(α′),

2sn = −1− N (vn(α)− wn(α) i) v′n(α′)− (v′n(α)− w′n(α) i) vn(α′)

N (vn(α) + wn(α) i) v′n(α′)− (v′n(α) + w′n(α) i) vn(α′).

Using the more common symbols for the Riccati-Bessel functions and through sim-ple rearrangement of the equations, we arrive at the Lorenz-Mie coefficients as theyappear in classic textbooks [van de Hulst 1957; Kerker 1969]:

−kn =ψn(α)ψ′n(α′)−Nψ′n(α)ψn(α′)

ζn(α)ψ′n(α′)−Nζ ′n(α)ψn(α′)= an ,

−sn =Nψn(α)ψ′n(α′)− ψ′n(α)ψn(α′)

Nζn(α)ψ′n(α′)− ζ ′n(α)ψn(α′)= bn ,

where the prime ′ denotes derivative when applied to one of the spherical functions,N is the relative index of refraction, and α and α′ are the size parameters (oftendenoted x and y, as mentioned in Section 6). The only differences between Lorenz’scoefficients and the modern ones an and bn are choice of symbols and choice of sign.Lorenz probably extracted −1 from the fractions to ease approximation in a timewithout computers.

Debye [1909a] reproduced the theory of Mie [1908] using scalar potentials, whichis now the method of choice in such derivations. These scalar potentials are todaycalled Debye potentials, but the same scalar potentials were used by Lorenz in 1890(as also mentioned by Kerker [1969] and in Section 6). Debye’s purpose was differentfrom Lorenz’s and Mie’s. He wanted to investigate the radiation pressure on smallspheres due to the momentum generated when an incident field is scattered or ab-sorbed. However, like Mie [1908], Debye only derived the expansion coefficients forthe scattered wave and neglected effects due to heat distribution inside the sphericalparticle. His theory therefore cannot explain photophoresis, which is the motion ofabsorbing particles in a fluid medium [Kerker 1982]. To consider the heat distributioninside a spherical particle and its influence on photophoretic force [Kerker and Cooke1973, 1982], the interior wave suddenly becomes of interest.

Some of the first generalizations of the Lorenz-Mie theory were to explain scat-tering by coated or multilayered (concentric) spheres [Aden and Kerker 1951; Kerker1969]. In such generalizations, the expansion coefficients change while the rest of thescattering theory remains more or less unchanged. The Lorenz-Mie coefficients thenserve as a limiting case where the coating and the particle have the same index ofrefraction. On the other hand, if we consider other heterogeneities inside the sphere,the interior wave again becomes of interest. This is the case, for example, if we wouldlike to study cell biology and consider molecular scattering [Chew et al. 1976] or adistribution of absorption centers [Dusel et al. 1979] within an irradiated sphere. Theexpansion coefficients of the interior wave (cn and dn) were derived by Kerker andCooke [1973] and in later work by other authors, often with an unnecessarily com-plicated numerator. Chew et al. [1976] found the simpler version of the numerator(Ni), but it is worth noting that these expansion coefficient were given by Lorenzin 1890 including the simple numerator [Lorenz 1890, Eqs. (34)]. Thus the completetreatment of the interior wave together with the scattered wave distinguishes Lorenz’swork from other early work on the scattering of a plane wave by a sphere.

An important generalization of the Lorenz-Mie theory is to consider an arbitraryposition of the scattering sphere in an arbitrarily shaped electromagnetic beam [Goues-bet and Grehan 2017]. The first version of this generalized Lorenz-Mie theory was


developed in the 1980s by Gouesbet and colleagues [Gouesbet and Grehan 1982;Gouesbet et al. 1988]. In this more general setting, one would use the original Lorenz-Mie coefficients multiplied by correction factors (same factors for an and cn and for bnand dn). Citing Dusel et al. [1979], the interior wave was introduced (without simplifi-cation) into the generalized theory by Barton et al. [1988]. The intensity distributionof the interior wave is required for explaining explosion characteristics of dropletsexposed to a very narrow laser beam [Schaub et al. 1989]. As another example, thisinternal intensity distribution is needed for reconstruction of the 3D temperaturefield within a combusting droplet [Castanet et al. 2005]. Gouesbet and Grehan [2017,Sec. 8.2] provide an overview of research employing generalized Lorenz-Mie theoryto calculate resonances in the internal field of a particle situated in a laser beam.They also explain (in Sec. 3.6) that it is possible to derive the simple numerator forthe Lorenz-Mie coefficients of the interior wave using a Wronskian relation. A factdiscovered by Lorenz in his work of 1890, but largely overlooked even today.

Another generalization of the Lorenz-Mie theory is to consider an absorbing hostmedium. As described by Mie [1908], Maxwell’s equations reveal that a nonzero con-ductivity leads directly to an imaginary part in the refractive index of the material,and this makes the material an absorbing medium. Early on, a derivation of theLorenz-Mie theory by Stratton [1941] showed that the Lorenz-Mie coefficients havethe same mathematical expressions regardless of the absorption/conduction prop-erties of the sphere and the medium. The only difference is whether the refractiveindices are real or complex numbers. This means that, regardless of the absorptionproperties of the particle and the host, the mathematical equations for computingthe Lorenz-Mie coefficients are as described by Lorenz in 1890. The output from theLorenz-Mie theory, however, can be quite different depending on whether the refrac-tive index of the host or the sphere is real-valued or complex-valued. The net effectof light scattering by particles thus changes depending on the absorption propertiesof the host medium and the particles.

Lorenz [1890] considered a transparent sphere in a transparent medium, which isvery useful for explaining atmospheric light scattering phenomena like sky, rainbows,and clouds. In this case, the refractive index of both particle and host are real numbers.Mie [1908] investigated an absorbing sphere in a transparent host to explain thecolourful scattering by colloidal metal solutions. An absorbing host is needed forexplaining the appearance of media like milk, icebergs, and natural waters [Frisvadet al. 2007b]. Milk serum and water and ice are all weakly absorbing host media inthese examples, like in many others. Light scattering by the atmosphere and by sea iceand seawater (and soot) are all highly important factors in climate models [Hansenand Nazarenko 2004] and modelling of global warming effects [Pitari et al. 2015].Wiscombe [1980] even wrote that “Mie scattering calculations pervade the entirefield of atmospheric optics.”

While the equations for the Lorenz-Mie coefficients are theoretically unaffectedby the absorption properties of the host medium, the distribution of the scatteredlight is not. This is not so important for polydisperse systems of particles in a weaklyabsorbing host [Frisvad et al. 2007b; Ma et al. 2019]. However, in cases like randomlasers [Luan et al. 2015] and solar cells [Kim et al. 2015], where the host is stronglyabsorbing and scattering by particles is used to improve efficiency, differences in thedistribution of the scattered light could be very important. Imagine for instance thatyour objective is to direct light into a photovoltaic cell using particle scattering.

The distribution of the scattered light changes when the incident wave becomesinhomogeneous [Frisvad 2018]. Interestingly, a plane wave becomes inhomogeneousupon refraction into an absorbing medium except in the unusual case of incidencealong the direction of the surface normal [Fry 1927]. Stratton did not consider aninhomogeneous wave. Early work on scattering by a sphere in an absorbing medium


follows Stratton [Galejs 1962; Mundy et al. 1974; Chylek 1977] and is thus limitedto the very rare case of a homogeneous wave in an absorbing medium. More recentwork [Belokopytov and Vasil’ev 2006], considers the case of an inhomogeneous wavein a transparent medium. The evanescent wave produced by the phenomenon of totalinternal reflection is an example of such a case. These waves are particularly importantin total internal reflectance microscopy [Prieve and Walz 1993], where the scatteringby small particles of an evanescent wave traveling along a flat surface is used todynamically detect and track the particles. The Lorenz-Mie coefficients (an, bn, cn,dn) are still the same as the ones found by Lorenz [1890], but it turns out thatan inhomogeneous wave is particularly effective at exciting the so-called whisperinggallery modes leading to microsphere resonators [Belokopytov and Vasil’ev 2006]. Ingeneral, the theory of scattering of an inhomogeneous wave by a sphere in an absorbingor a transparent host is not so far from the original Lorenz-Mie theory [Frisvad 2018].However, the rotational symmetry of the distribution of the scattered light aroundthe direction of the incident light is broken by the wave inhomogeneity.

Microsphere resonators have a surprising number of interesting applications inbiosensing, optical signal processing, and development of microlasers [Rakovich andDonegan 2010; Ward and Benson 2011]. The Lorenz-Mie coefficients are directly usedto calculate the conditions for an optical resonance [Rakovich and Donegan 2010].Particles may even be trapped in a resonant cavity. This phenomenon of a radia-tion pressure laser trap with the ability to levitate a particle is referred to as opticaltweezers. In this context, the internal field distribution is again important, and wecan calculate this for a particle in a laser beam using generalized Lorenz-Mie the-ory [Gouesbet 2019]. Optical tweezers have a host of applications and even spurredthe rise of single-molecule biophysics [Killian et al. 2018; Polimeno et al. 2018]. The fa-ther of optical tweezers, Arthur Ashkin, was awarded the 2018 Nobel prize in physics.

It is interesting to ponder whether Lorenz’s unique theoretical analysis of the in-ternal field carries some useful insights just waiting to be collected. In particular,Lorenz’s practical approximate formulae for the part of the interior wave due to totalinternal reflection seem highly interesting [Lorenz 1890, p. 52]. In a geometrical opticsexplanation, these resonances referred to as whispering gallery modes are due to lightbeing trapped inside the particle by total internal reflection. An approximate empir-ical formula is often used in the analysis of such whispering gallery modes [Rakovichand Donegan 2010]. Lorenz’s formulae are self-contained without series expansions,and like the empirical formula they involve an inverse tangent function (in the vari-ables δ and ∆). Thus, they could perhaps lead to a new practical but theoreticallyfounded formula for analyzing the structure of whispering gallery modes.

9 Lorenz-Mie theory and appearance prediction

The Lorenz-Mie theory is particularly useful for computing the macroscopic scatteringproperties of a medium containing a random distribution of scattering particles, aso-called random medium. The seminal textbook in the area of wave propagationin random media is that of Ishimaru [1978]. We note that Ishimaru in his latestbook describes “the Mie scattering of a dielectric sphere” in more detail and nowmentions that “Lorenz gave essentially the same results before Mie’s work” [Ishimaru2017, p. 382]. This means that Lorenz’s contribution in all fairness is becoming morewidely recognized. The case of dielectric spheres is also precisely the one addressed byLorenz, while Mie had his focus on conducting spheres. The macroscopic scatteringproperties would be the phase function p and the scattering coefficient µs (which isthe same as h for non-absorbing host and particles). The phase function is the farfield distribution of the scattered light, while the scattering coefficient is the amount


of scattering per unit of distance that light has travelled through the medium. Oncesuch properties are available, we can use them as parameters in the radiative transferequation [Chandrasekhar 1950; Ishimaru 1978], which together with a theory of lightpropagation (often geometrical optics and thus ray tracing) forms the basis of materialappearance prediction [Frisvad et al. 2007b, 2012].

The link provided by Lorenz-Mie theory between the particle composition of arandom medium and its scattering properties and appearance in an image has abroad range of applications. These are not only in atmospheric optics (as mentionedabove), but also in biomedical optics [Wang and Wu 2007; Tuchin 2015], hydrologicaloptics [Mobley 1994], heat transfer [Howell et al. 2016], and computer graphics [Callet1996; Frisvad et al. 2007b]. The Lorenz-Mie theory is prominently featured in seminaltextbooks of all these areas except the last. The link from particle composition toappearance becomes important as soon as we are interested in analysis based onimaging modalities. As an example, we can with a noninvasive hyperspectral imageof a laser spot reflected from a random medium use Lorenz-Mie theory to estimate sizedistributions of one or two particle inclusions [Abildgaard et al. 2016]. This inverseuse of Lorenz-Mie theory has been around at least since the mid-90s [Box et al. 1992;Mourant et al. 1997], and a recently conducted sensitivity analysis found a goodpotential for particle sizing in this approach [Postelmans et al. 2018].

Lorenz [1890, p. 56] provides the equations used today for calculating the phasefunction and the scattering cross section of a spherical particle. The phase functionis provided in terms of the transversal components of the wave vector:

Eφ = −Hθ = ζe =i sinφ

krei(ωt−kr)

∞∑n=1

2n+ 1

n(n+ 1)

(kn

dPnsin θ dθ

+ snd2Pndθ2

)

Eθ = Hφ = ηe = − i cosφ

krei(ωt−kr)

∞∑n=1

2n+ 1

n(n+ 1)

(knd2Pndθ2

+ sndPn

sin θ dθ

).

Lorenz’s result is equivalent to the transversal components of the scattered field vec-tors as found in textbooks deriving the theory based on Maxwell’s equations [van deHulst 1957; Kerker 1969]. The previously mentioned sign difference in Lorenz’s expan-sion coefficients is canceled by Lorenz not using the first order associated Legendrepolynomials, since P 1

n = −dPn

dθ . For the scattering cross section, Lorenz found

Cs =λ2

2π

∞∑n=1

(2n+ 1)(|kn|2 + |sn|2) ,

which is also exactly the same as in modern textbooks given that kn = −an andsn = −bn. Based on Lorenz’s exposition, the macroscopic scattering properties of amonodisperse medium with one type of particle of one size are then

p(θ) =|S1(θ)|2 + |S2(θ)|2

2|k|2Cs

S1(θ) =

∞∑n=1

2n+ 1

n(n+ 1)

(kn

dPnsin θ dθ

+ snd2Pndθ2

)

S2(θ) =

∞∑n=1

2n+ 1

n(n+ 1)

(knd2Pndθ2

+ sndPn

sin θ dθ

)µs = h = NpCs .

Remarkably, Lorenz uses these macroscopic scattering properties for precisely theinverse problem discussed above. Toward the end of his memoir, he states the follow-ing [Lorenz 1890, pp. 60–61]:


If one has determined wavelengths of a series of absorption lines for a sys-tem, these can be ascribed to the reciprocals of the roots in vn(β) = 0,n = 0, 1, 2, . . ., through multiplication by a single constant factor. As thisfactor is equal to N

2πR , it will thus be possible from the refractive index ofthe system and from its ordinary absorption coefficient to determine all theconstants of the system, namely the number of spheres in a unit of volume,the size of the spheres and their refractive index.

Lorenz’s suggested approach, which is described in more detail in the concludingpages of the memoir, could easily carry some advantages over the more uninformeddata fitting approaches employed in more recent papers.

According to Christiansen [1896] who wrote about Lorenz only a few years afterthe 1890 treatise, Lorenz had set himself the goal to fully solve the old famous rainbowproblem. Thus, Lorenz spent a number of pages on appearance predictions regardingthe rainbow (especially pp. 38–42 of the original publication). As an example, Lorenzprovided the following formula [Lorenz 1890, p. 42]:

Max. of apparent brightness = 0.067281

π

(R

λ

) 13

.

Suppose we observe a rainbow, then given the relation between wavelengths andperceived colours [Helmholtz 1867], the brightness of the observed colours over thebackground would reveal the mean radius of the raindrops forming the rainbow.Lorenz cited Helmholtz in his work on dispersion [Lorenz 1883] and was certainlyaware of the link between wavelength and perceived colour. The Young-Helmholtztheory of trichromatic colour vision defines three curves that specify the perceptionof a primary colour as a function of wavelength. The three primary colours are red,green, and blue, and they roughly correspond to the three colour sensitive cones in thehuman eye. Using psychophysical measurements of these curves [Stiles and Burche1959; Stockman and Sharpe 2000], we can produce a table of the colours correspondingto different wavelengths. This can then be used together with Lorenz’s theory.

Figure 3 is an example of an inverse method where we use Lorenz’s rainbowtheory for analysis of waterdrop size distributions in rainbows. In addition, Lorenzprovided equations for calculating the apparent brightness of the individual spectralcolours [Lorenz 1890, Eqs. (a), (b’), and (c’)]. This corresponds to appearance pre-diction. Interestingly, computer graphics today enables us to test the visual qualityof historical theories by rendering images based on a given theory [Frisvad et al.2007a]. We can for example test Aristotle’s rainbow theory, which is based on re-flecting spherical particles. This can produce a primary rainbow effect at very lowcomputational expense [Frisvad et al. 2007a], but the theory accounts for no specificsin the appearance of rainbows. The visual output of Lorenz’s theory has been testedfor atmospheric phenomena (sky and rainbow) by Jackel and Walter [1997] and indetail for rainbows by Sadeghi et al. [2012]. In this detailed investigation, the au-thors found that Lorenz’s theory explains most visual rainbow effects. However, forlarger waterdrops (R > 0.4 mm), modification is required to account for some visualphenomena as the perfect sphere is no longer as good an approximation.

The ability of the Lorenz-Mie theory to serve as a tool for appearance predic-tion has surprising quantitative accuracy despite the fact that many particles are notperfect spheres. Surface tension, which forces small liquid particles toward a morespherical shape, is probably one of the reasons behind the success of the theory.When particles are not quite spheres, it mostly affects the accuracy of the directionaldistribution of the scattered light (the phase function p). However, in a polydispersecolloid with particles of many different sizes, this is not quite as important as when


390 nm 560 nm 730 nm

Fig. 3. Two photographs of rainbows and the spectral colours corresponding to visiblewavelengths. The background shines through for the wavelengths where the human eye isnot very sensitive. We pick pixel samples (dots in images) along a line crossing the rainbowand estimate wavelength and apparent brightness of the scattered light for each pixel bycomparing with the spectral colours. Using Lorenz’s theory, we find mean waterdrop radiusof µ = 0.26 mm (σ = 0.12 mm) for the left photo and µ = 0.35 mm (σ = 0.35 mm) for theright photo. These are plausible values corresponding well with the fact that the brightestrainbow pixels in the left image are green while they are red in the right image.

we consider an engineered material like a solar cell. In polydisperse colloids, an ap-proximate phase function based on the asymmetry parameter g (the mean cosineof the scattering angle) is often sufficient. In case of non-spherical particles, we canobtain fairly accurate scattering coefficient (µs) and asymmetry parameter (g) usingLorenz-Mie theory on a collection of spheres with volume-to-surface-area ratio thatis equivalent to that of the non-spherical particles [Grenfell and Warren 1999]. Thesetwo scattering properties, including their wavelength dependence, are critical for thequantitative accuracy of appearance predictions.

Conclusively, we illustrate the quantitative accuracy of the Lorenz-Mie theorywhen predicting the appearance of a cloudy beverage in Figure 4. As seen in the com-parison of the photographed image and the predicted image, appearance differencesare mostly a consequence of slight deviations in the glass geometry and reflectance,and in the reflectance of the background made of white cardboard. The colour andtranslucency of the cloudy beverage are predicted quite convincingly. The advantageof such a model is that we can modify particle size distributions and absorption spec-tra based on production parameters. In this way, the Lorenz-Mie theory can helppredicting the consequences of modified production parameters with respect to prod-uct appearance. In addition, we can use calibrated photography (computer vision) tocheck whether product samples are likely to be according to specification. Lorenz-Mietheory could thus become an important tool in the digitalization of quality assuranceand fault detection that we see in industry these years.

10 Conclusion

We firmly believe that the Lorenz-Mie theory will persist for many years to come. Inmany ways, Lorenz’s memoir seems a lost treasure. It is certainly a mathematicallydense text, but on the other hand it explains mathematical details that it could takea long time to unearth for the uninitiated. We believe that closer scrutiny of the equa-tions presented by Lorenz could be a fruitful endeavour. In the preceding sections,


Light: Bowens BW3370 100W Unilite (6400K)

DLSR camera, 50 mm lens

cloudy beverage

Backdrop: white cardboard

Fine cloudCoarse cloud

µ

black measurement red model

wavelength [nm]400 450 500 550 600 650 700 750

imag

inar

y pa

rt o

f IO

R

×10-5

0

0.2

0.4

0.6

0.8

1

1.2

filtered apple juice (real part of IOR: 1.35)apple flesh (real part of IOR: 1.487)

photograph rendering

Fig. 4. Let us consider the physical experiment of acquiring a photograph of a polydispersecolloid with a digital camera. The experimental setup is sketched in the upper left corner.We choose cloudy apple juice as our colloid and use an available appearance model basedon Lorenz-Mie theory [Dal Corso et al. 2016]. The model produces a bimodal particle sizedistribution based on the weight-% of colloidal apple flesh particles (lower left plot, D isparticle diameter) and index of refraction (IOR) for the apple juice host and the particles(lower right plot). The appearance predicted by this model is presented in the upper rightimage. Below photograph and rendering, we have extracted a small part of each image andpresent a third image: the absolute difference in each colour band multiplied by 10.

we have tried to hint at different parts of the text that might be of interest. Lorenz’sdetailed theoretical investigation of the internal field might be useful with respectto formulation of new practical ways of finding internal resonances in the scattering.The many pages dedicated to approximations could perhaps lead to ingenious inversemethods that we might not otherwise have thought of. In any case, we find it perti-nent that Lorenz’s 1890 treatise with our translation to English is now more broadlyavailable to the scientific community.

Acknowledgements. The picture in Fig. 1 is courtesy of the Royal Library of Copen-hagen and included with permission for which we are grateful. The rainbow pho-tographs of Fig. 3 were taken by the first author respectively in Maui, Hawaii (left),and Bagsværd, Denmark (right). The reference photograph in Fig. 4 also appearedin Dal Corso et al. [2016], but copyright was retained by the authors and not trans-ferred for this publication (and the first author of the present work was one of theseauthors).


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